广义(2+1)维Boussinesq方程的孤立波解

首先应用Riccati展开法获得广义(2+1)维Boussinesq方程的96组相互作用解,这类解同时含有三角函数、双曲函数、有理函数、指数函数等,它反映了不同类型非线性波的相互作用.然后应用同宿测试方法结合Hirota双线性形式求得广义(2+1)维Boussinesq方程的周期孤波解,通过相应的时空变换,得到方程其他形式的解.     

解非线性椭圆型方程奇异摄动问题的一致收敛差分格式

给出了求解非线性椭圆型偏微分方程奇异摄动问题的广义OCI差分格式．证明了这种格式的解关于摄动参数一致收敛于连续问题的解．给出了数值例子．     

应用Riccati-Bernoulli辅助方程求解广义非线性Schr?dinger方程和(2+1)维非线性Ginzburg-Landau方程

研究了Riccati-Bernoulli辅助方程法,并应用这种方法得到广义非线性Schr?dinger方程和(2+1)维非线性Ginzburg-Landau方程的精确行波解.这些解包括有理函数、三角函数、双曲函数和指数函数.应用这种方法求解过程简洁有效.该研究对于数学物理方程领域诸多非线性偏微分方程精确解的探究具有重要的意义.     

有限变形弹性圆杆中的孤波

在同时引入横向惯性和横向剪切应变的情况下,导出了有限变形弹性圆杆的非线性纵向波动方程,方程中包含了二次和三次的非线性项以及由横向剪切与横向惯性导致的两种几何弥散效应．借助Mathematica软件,利用双曲正割函数的有限展开法,对该方程和对应的截断的非线性方程进行求解,得到了非线性波动方程的孤波解,同时给出了这些解存在的必要条件．     

矩形网格扁壳结构的非线性振动

本文运用作者已建立的矩形网格扁壳的非线性弹性理论,求解了该类结构的非线性振动问题。通过采用横向挠度（网格节点横向位移）和力函数的某种（广义）Fourier级数形式的设定解,由试函数的加权得到解中系数之间的关系和决定时间未知函数的振动方程,然后利用正则摄动法和迦辽金法推导出结构自由振动和谐和激励作用下结构非线性受迫振动的幅频关系,并给出了计算实例。     

一类非线性双曲型发展方程的孤子解

研究了一类非线性发展方程.首先在无扰动情形下,利用待定函数和泛函同伦映射方法得到了非扰动发展方程的孤子精确解和扰动方程的任意次近似行波孤子解.接着引入一个同伦映射,并选取初始近似函数,再用同伦映射理论,依次求出非线性双曲型发展扰动方程孤子解的各次近似解析解.再利用摄动理论举例说明了用该方法得到的近似解析解的有效性和各次近似解的近似度.最后,简述了用同伦映射方法得到的近似解的意义,指出了用上述方法得到的各次近似解具有便于求解、精度高等优点.     

一类具波动算子非线性Schr?dinger方程的精确解

研究一类具波动算子非线性Schr?dinger方程的精确解问题.引入Jacobi椭圆函数组合及双曲函数组合方法,将其应用于求解具有波动算子的非线性Schr?dinger方程中.通过简单代数运算,可以得到具有波动算子非线性Schr?dinger方程的许多新解,并在极限情况下,给出了该方程对应的双曲函数解.同时得出了双曲函数组合解是Jacobi椭圆函数组合解情况下的极限解的结论.该方法可以推广到更多非线性偏微分方程精确解求解问题.     

关于一类非线性双曲方程组的二维Cauchy问题

本文通过引进新的广义解定义，对一类非线性双曲方程组的二维Ｃａｕｃｈｙ问题，证明了解的存在唯一性．并且，解可能含δ波．     

几种广义非线性发展方程的新解

本文研究了构造了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解的问题.利用三种辅助方程及其新解,获得了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解.这些解由双曲余割函数、双曲正切函数、双曲正割函数、双曲余切函数和余割函数组成.     

(3+1)维Klein-Gordon方程的新求解方法

给出第一种椭圆方程与函数变换相结合的方法,通过几个步骤,构造了(3+1)维Klein-Gordon方程的多种新解.步骤一、根据Jacobi椭圆函数的性质,获得了第一种椭圆方程的几种新解.步骤二、用第一种椭圆方程与函数变换相结合的方法,将(3+1)维Klein-Gordon方程的求解问题转化为非线性代数方程的求解问题.步骤三、借助符号计算系统Mathematica求出该方程组的解,并构造了由Riemannθ函数、Jacobi椭圆函数、双曲函数和三角函数两两组合的双周期解和双孤子解等多种复合型新解.     

A new method for homoclinic solutions of ordinary differential equations

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int J Non-Linear Mech 1999;34(4):759–78; Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;147(2):499–513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006;360:109–13; Hayat T, Sajid M. On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function can be obtained as a special case of this homotopy analysis method. Finally, homoclinic solutions are obtained (for nontrivial examples) analytically by HAM, and are presented through graphs.     

A new modification of He’s homotopy perturbation method for rapid convergence of nonlinear undamped oscillators

In this paper we present a new efficient modification of the homotopy perturbation method with x ³ force nonlinear undamped oscillators for the first time that will accurate and facilitate the calculations. The He’s homotopy perturbation method is modified by adding a term to linear operator depends on the equation and boundary conditions. We find that this modified homotopy perturbation method works very well for the wide range of time and boundary conditions for nonlinear oscillator. Only two or three iteration leads to high accuracy of the solutions. We then conduct a comparative study between the new modification and the homotopy perturbation method for strongly nonlinear oscillators. Numerical illustrations are investigated to show the accurate of the techniques. The new modified method accelerates the rapid convergence of the solution, reduces the error solution and increases the validity range. The new modification introduces a promising tool for many nonlinear problems.     

Prediction of homoclinic bifurcation: the elliptic averaging method

A criterion to predict bifurcation of homoclinic orbits in strongly nonlinear autonomous oscillators is presented. The averaging method combined formally with the Jacobian elliptic functions is applied to determine an approximation of limit cycles near homoclinicity. We then introduce a criterion for predicting homoclinic orbits, based on the collision between the bifurcating limit cycle and the saddle equillibrium. In particular, we show that this criterion leads to the same results as the standard Melnikov technique. Explicit applications of this criterion to quadratic nonlinearities are included.     

Approximate period calculation for some strongly nonlinear oscillation by He’s parameter-expanding methods

In this paper, He’s modified Lindstedt–Poincare method and bookkeeping parameter method, also known as He’s parameter-expanding method, is applied to various kinds of strongly nonlinear oscillators. We obtained sufficiently accurate solutions with one iteration which is valid for whole domain to the contrary of classical perturbation techniques.     

无小参数系统的混沌与亚谐共振

Melnikov方法是判别混沌和亚谐共振的一种重要方法．传统的Melnikov方法依赖于小参数,在大多数实际物理系统中,小参数是不存在的．因此,传统的Melnikov方法不能应用于强非线性系统．为了摆脱小参数对Melnikov方法的限制,采用同伦分析将Melnikov方法拓展到强非线性系统,且采用该方法研究了一个强非线性系统的亚谐共振与混沌,解析结果和数值结果相互吻合,说明了该方法的有效性．     

Abundant explicit exact solutions to the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law nonlinearities

This paper is concerned with the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law. Abundant explicit and exact solutions of the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law are derived uniformly by using the first integral method. These exact solutions are include that of extended hyperbolic function solutions, periodic wave solutions of triangle functions type, exponential form solution, and complex hyperbolic trigonometric function solutions and so on. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial DEs. Copyright © 2014 John Wiley & Sons, Ltd.     

On the rational solitary wave solutions for the nonlinear Hirota–Satsuma coupled KdV system

The objective of this article is to investigate an algebraic method for constructing new rational exact wave soliton solutions in terms of hyperbolic and triangular functions for the generalized nonlinear Hirota–Satsuma coupled KdV systems of partial differential equations using symbolic software like Mathematica or Maple. These studies reveal that the generalized nonlinear Hirota–Satsuma coupled KdV system has a rich variety of solutions.     

On the existence of generalized solutions and the convergence of difference methods for nonlinear initial-value problems

In the present paper we prove new results for a general perturbation theory for nonlinear mappings between metric spaces. Using these results we are able to establish new principles for the treatment of nonlinear initial-value problems by difference methods. The main results are the characterization of the existence of discrete limits of sequences of mappings and the characterization of the existence of generalized solutions of nonlinear initial-value problems which are limits of solutions of difference equations. As conclusions one obtains generalizations of Lax's equivalence theorem for nonlinear and linear initial-value problems and a convergence theorem for a concrete hyperbolic equation.     

Generalization of a Relaxation Scheme for Systems of Forced Nonlinear Hyperbolic Conservation Laws with Spatially Dependent Flux Functions

A generalization of a finite difference method for calculating numerical solutions to systems of nonlinear hyperbolic conservation laws in one spatial variable is investigated. A previously developed numerical technique called the relaxation method is modified from its initial application to solve initial value problems for systems of nonlinear hyperbolic conservation laws. The relaxation method is generalized in three ways herein to include problems involving any combination of the following factors: systems of nonlinear hyperbolic conservation laws with spatially dependent flux functions, nonzero forcing terms, and correctly posed boundary values. An initial value problem for the forced inviscid Burgers' equation is used as an example to show excellent agreement between theoretical solutions and numerical calculations. An initial boundary value problem consisting of a system of four partial differential equations based on the two-layer shallow-water equations is solved numerically to display a more general applicability of the method than was previously known.     

New kinds of solitons and periodic solutions to the generalized KdV equation

In this work, the sine‐cosine method, the tanh method, and specific schemes that involve hyperbolic functions are used to study solitons and periodic solutions governed by the generalized KdV equation. New solutions are determined by using the hyperbolic functions schemes. The study introduces new approaches to handle nonlinear PDEs in the solitary wave theory. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 247–255, 2007